Question:

<p><strong>Step 1:</strong> Decompose the integrand into odd and even parts.</p><p>Let f(x) = x + x². Here x is an odd function and x² is an even function.</p><p><strong>Step 2:</strong> Apply the property that the integral of an odd function over a symmetric interval is zero.</p><p>$$\int_{-\pi}^{\pi} x \, dx = 0$$</p><p><strong>Step 3:</strong> Compute the integral of the even function x².</p><p>$$\int_{-\pi}^{\pi} x^2 \, dx = 2\int_{0}^{\pi} x^2 \, dx = 2\left[\frac{x^3}{3}\right]_0^{\pi} = 2 \cdot \frac{\pi^3}{3} = \frac{2\pi^3}{3}$$</p><p><strong>Step 4:</strong> Calculate a₀.</p><p>$$a_0 = \frac{1}{\pi}\int_{-\pi}^{\pi}(x + x^2)\,dx = \frac{1}{\pi}\left(0 + \frac{2\pi^3}{3}\right) = \frac{2\pi^2}{3}$$</p><p>∴ Answer: A</p>